*(note: this post has been closed to comments; comments about it on other pages will be deleted!)*

I really didn't think this would go this far. I've gotten thousands of hits in the last couple of days, with links to the original post showing up all over the place on link farms. I will post a longer response in a couple of day, but I don't have time right now because I'm on vacation.

I will, of course, continue to believe my original claim, and thanks to the math people who post to help me explain it. I will not be able to answer every single comment (I can't even keep up with reading them). I hope that some of the people who read the post, though, were enlightened at least a little bit by the formal view of this question (I know I didn't include all the formalities, but I wasn't writing for people that would want them; the formalities are there, but I'm not going to try to explain them. This wikipedia article does a good job, if you're interested).

Back in a few days....

I view this as an artifact of our use of base 10. So, if we were to use base three, 1/3(in decimal) is 0.1. And in base 3,

0.1+0.1+0.1=1

That said, you get the even more counter-intuitive proposition: 0.222222... = 1 in base 3.

I think the whole problem comes out as there is no way to represent the number closest, but still less than 1. The number exists, but our methods of representing numbers is incomplete.

Posted by: Cober | September 30, 2007 at 09:38 PM

Why would someone think that .9 no matter what else comes after would magically change to 1.0? .9 is .9 and even an infinite amount of 9s will not change the value of that first 9!!! Therefore .999 does not equal one, it is just close. And can an infinite amount of 9s or anything for that matter even exist?

Posted by: CM | March 16, 2008 at 11:35 PM

I'd like to point out a rather fundamental flaw with your "proof".

Let x=1 and y=.9999...

Now I need to remind you of a basic identity property that any number minus "itself" equals 0.

SO, x-y 0

In your older post you said the following:

"Let me guess: the average is .99999...05? So after this infinite list of 9s, there's the possibility of starting up multiple-digit extensions? Doesn't that just raise the obvious question: What about .9999999...9999999...? Namely, infinitely many 9s, and then after that infinite list, there's another infinite list of 9s? How, exactly is that different from the original infinite list of 9s? If you saw it written out, where would the break between the lists be?"

The same applies to .000....000... The fact is that it IS infinite, but there is something at the end. And infinitely small number does not equal one and further more, adding this infinitely small number to 1 would make it greater than one, but adding it to .999... would make it 1! You cannot argue that it does not exist due to the same logic you are using to say that it is infinitely close to 1. Close does not mean equal.

Posted by: Matt | May 11, 2008 at 06:19 PM

Matt, what is this "something at the end" you speak of? You do not say what it is, nor do you do any calculations to demonstrate the properties you assign to it. You provide no evidence whatsoever that it exists at all. In other words, you're just like that chicken who runs around claiming that the sky is falling.

I suggest you go to Wikipedia and read the articles there about "Real Numbers" and "0.999...".

Also, if you deign to reply at all, show me your calculation of "1 - 0.999... =". And no, you are not allowed to replace 0.999... with "0.999...9" or similar. The moment you do so, you admit that you have to lie and cheat to prove your case.

Posted by: Monimonika | May 12, 2008 at 10:17 AM